Periods of automorphic forms

Authors:
Hervé Jacquet, Erez Lapid and Jonathan Rogawski

Journal:
J. Amer. Math. Soc. **12** (1999), 173-240

MSC (1991):
Primary 11F55, 11F70, 11F72

DOI:
https://doi.org/10.1090/S0894-0347-99-00279-9

MathSciNet review:
1625060

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a quadratic extension of number fields and , where is a reductive group over . We define the integral (in general, non-convergent) of an automorphic form on over via regularization. This regularized integral is used to derive a formula for the integral over of a truncated Eisenstein series on . More explicit results are obtained in the case . These results will find applications in the expansion of the spectral side of the relative trace formula.

**[A1]**J. Arthur,*A trace formula for reductive groups I*, Duke Math. J.,**45**, No. 1 (1978), 911-952. MR**80d:10043****[A2]**J. Arthur,*A trace formula for reductive groups II*, Compositio Math.,**40**(1980), 87-121. MR**81b:22018****[A3]**J. Arthur,*The trace formula in invariant form*, Duke Math. J.,**114**(1981), 1-74. MR**84a:10031****[A4]**J. Arthur,*On the inner product of truncated Eisenstein series*, Duke Math. J.,**49**(1982), 35-70. MR**83e:22023****[AS]**T. Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin's method, Math. Ann.**266**(1977), 81-94. MR**55:2761****[BZ]**I. Bernstein and A. Zelevinsky,*Induced representations of reductive**-adic groups I*, Ann. scient. Éc. Norm. Sup., 4 série, t.**10**, 1977, p. 441-472. MR**58:28310****[BR]**D. Blasius and J. Rogawski,*Cohomology of congruence subgroups of**and Hodge cycles on some special complex hyperbolic surfaces*, to appear in: Proc. of conference on Regulators, Jerusalem 1996 (Ed. A. Resnikov and N. Schappacher).**[C]**W. Casselman,*Extended automorphic forms on the upper half-plane*, Math. Annalen**296**(1993), 755-762. MR**94k:11049****[F1]**Y. Flicker,*Twisted Tensor and Euler Products*, Bull. Soc. Math. France,**52**(1989), 295-313. MR**89m:11049****[F2]**Y. Flicker,*On distinguished representations*, J. Reine Angew. Math.**418**(1991), 139-172. MR**92i:22019****[F3]**Y. Flicker,*Distinguished Representations and a Fourier Trace Formula*, Bull. Soc. Math. France,**120**(1992), 413-465. MR**93j:22033****[F4]**Y. Flicker,*Cyclic automorphic forms on a unitary group*, J. Math. Kyoto Univ.**37**(1997), 367-439. CMP**98:07****[FH]**Y. Flicker and J. Hakim,*Quaternionic Distinguished Representations*, American J. Math.,**116**(1994), 255-278. MR**95i:22028****[FJ]**S. Friedberg and H. Jacquet,*Linear periods*, J. Reine Angew. Math.**443**(1993), 91-139. MR**94k:11058****[FGJR]**S. Friedberg, S. Gelbart, H. Jacquet, and J. Rogawski,*On genericity for -packets*, pre-print.**[GJR]**S. Gelbart, H. Jacquet, and J. Rogawski,*The relative trace formula and genericity of L-packets for*In preparation.**[GRS]**S. Gelbart, D. Soudry, and J. Rogawski,*Endoscopy, theta-liftings, and period integrals for the unitary group in three variables*Annals of Math.**145**(1997), 1-58. CMP**97:14****[GU]**J. Guo,*On the positivity of central critical values of automorphic L-functions of GL(2)*, Duke Math. J.,**83**(1996), 157-190. MR**97h:11055****[G]**N. Gurevitch, Unpublished notes.**[HLR]**G. Harder, R. Langlands, and M. Rapoport,*Algebraische zyklen auf Hilbert-Blumenthal Flächen*, J. reine angew. Math.**366**(1986), 53-120. MR**87k:11066****[J1]**H. Jacquet,*Sur un résultat de Waldspurger*, Ann. Scient. Ec. Norm. Sup.,**19**(1986), 185-229. MR**88d:11051****[J2]**H. Jacquet,*The continuous spectrum of the relative trace formula for**over a quadratic extension*, Israel J. Math.,**89**(1995), 1-59. MR**96a:22029****[J3]**H. Jacquet,*Automorphic spectrum of symmetric spaces*, In:*Representation Theory and Automorphic Forms*, (Eds. T. N. Bailey and A. W. Knapp) Proc. Symposia in Pure Math.**61**, AMS, Providence, 1997. CMP**98:03****[JLR]**H. Jacquet, K. Lai, S. Rallis,*A trace formula for symmetric spaces*, Duke Math. J., , No. 2 (1993), 305-372. MR**94d:11033****[JL]**H. Jacquet and K. Lai,*A relative trace formula*, Comp. Math.,**54**(1985), 243-310. MR**86j:11059****[JR]**H. Jacquet and S. Rallis,*Symplectic periods*, J. Reine Angew. Math.,**423**(1992), 175-197. MR**93b:22035****[JS]**H. Jacquet and J. Shalika,*On Euler products and the classification of automorphic representations I,II*, Amer. J. Math.**103**(1981), 499-558 and 777-815. MR**82m:10050a**; MR**82m:10050b****[JY]**H. Jacquet and Y. Ye,*Une remarque sur le changement de base quadratique*, C.R. Acad. Sci. Paris Sér. I , Math.**311**(1990), 671-676. MR**92j:11046****[JI]**D. Jiang,*Periods of residual representations of*, pre-print.**[LA1]**K. F. Lai,*Algebraic Cycles on Compact Shimura Surface*, Math. Z.**189**(1985), 593-602. MR**87a:11057****[LA2]**K. F. Lai,*On Arthur's Class Expansion of the Relative Trace Formula*, Duke Math. J.**64**(1991), 111-117. MR**92k:22031****[L]**R. Langlands,*Eisenstein series*, in*Algebraic Groups and Discontinuous Subgroups,*Proc. Symp. Pure Math. vol.**9**, AMS 1966. MR**40:2784****[LP]**E. Lapid,*Thesis*, Weizmann Institute of Science, 1998.**[LE]**J. Levy,*A note on the relative trace formula*, Canad. Math. Bull. vol.**38**(1995), 450-461. MR**97d:11085****[LR]**G. Laumon and M. Rapoport,*The Langlands lemma and the Betti numbers of stacks of**-bundles on a curve*, Internat. J. Math.**7**(1996), 29-45. MR**97f:14012****[MA]**Z. Mao,*Relative Kloosterman integrals for the unitary group*, C.R. Acad. Sci. Paris Sér. I Math 315 (1992), 381-386. MR**93g:11051****[MW]**C. Moeglin and J.-L. Waldspurger,*Decomposition spectrale et series D'Eisenstein*, Birkhauser 1993. MR**95d:11067****[SA]**P. Sarnak,*Diophantine problems and linear groups*, In:*Proceedings of Int. Cong. Math.*, Kyoto 1990, Springer Verlag 1991, 459-471. MR**93g:11054****[S]**T. Springer,*Some results on algebraic groups with involutions*, in*Algebraic groups and related topics,*Advanced Studies in Pure Mathematics, vol.**6**, 1985, 525-543. MR**86m:20050****[Z]**D. Zagier,*The Rankin-Selberg method for automorphic functions which are not of rapid decay*, J. Fac. Sci. Univ. Tokyo (1982), 415-437. MR**83k:10056**

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Additional Information

**Hervé Jacquet**

Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027

Email:
hj@math.columbia.edu

**Erez Lapid**

Affiliation:
Department of Mathematics, Weizmann Institute, Rehovot, Israel

Email:
erezl@wisdom.weizmann.ac.il

**Jonathan Rogawski**

Affiliation:
Department of Mathematics, Hebrew University, Jerusalem, Israel

Address at time of publication:
Department of Mathematics, University of California, Los Angeles, California 90095

Email:
jonr@math.huji.ac.il

DOI:
https://doi.org/10.1090/S0894-0347-99-00279-9

Received by editor(s):
October 14, 1997

Received by editor(s) in revised form:
May 14, 1998

Additional Notes:
The first author was partially supported by NSF Grant 9619766.

The third author was partially supported by NSF Grant 9401466.

Article copyright:
© Copyright 1999
American Mathematical Society